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temata:17a-matematicka_analyza:diferencialni_rovnice [2011/03/03 09:38] vagabund |
temata:17a-matematicka_analyza:diferencialni_rovnice [2011/04/21 11:46] (aktuální) vagabund |
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|---|---|---|---|
| Řádek 5: | Řádek 5: | ||
| ==== Názvosloví ==== | ==== Názvosloví ==== | ||
| + | <box round 90% blue|**Definice**> | ||
| + | **Diferenciální rovnice** - každá rovnice obsahující derivace (obyčejné, parciální), | ||
| + | * př. <m>ay\prime\prime + by\prime + cy = g(t)</m> | ||
| + | * př. <m>sin(y){d^2y}/{dx^2} = (1 - y){dy}/{dx} + y^2e^{-5y}</m> | ||
| + | * př. <m>{\partial^3u}/{\partial^2x\partial t} = 1 + {\partial u}/{\partial y}</m> | ||
| + | |||
| + | **Řád derivace** - nejvyšší derivace obsažená v rovnici | ||
| + | |||
| + | **Obyčejné(ODE)/Parciální(PDE) diferenciální rovnice** - obsahující derivace podle jedné/více proměnných | ||
| + | |||
| + | **Lineární diferenciální rovnice** - každá rovnice tvaru <m>a_n(t)y^{(n)}(t) + a_{n - 1}(t)y^{(n - 1)}(t) + ... + a_1(t)y\prime(t) + a_0(t)y(t) = g(t)</m> | ||
| + | |||
| + | **Řešení na intervalu J** - každá fce, pro kterou je na J splněna daná rovnice | ||
| + | |||
| + | **Počáteční podmínky (IV)** - jednoznačně určují konkrétní řešení dané rovnice | ||
| + | |||
| + | **IVP - Initial Value Problem** - DE s počátečními podmínkami | ||
| + | |||
| + | * př. <m>2ty\prime + 4y = 3, y(1) = -4</m> | ||
| + | |||
| + | **Interval of Validity for IVP** - největší možný interval, na kterém je dané řešení rovnice validní | ||
| + | |||
| + | **Obecné řešení (General Solution)** - Řešení bez počátečních podmínek | ||
| + | |||
| + | **Actual Solution** - GS s dosazením IV | ||
| + | |||
| + | **Explicitní/Implicitní řešení** - y = f(t)/0 = f(y,t) | ||
| + | |||
| + | </box> | ||
| + | |||
| + | === Diferenciální rovnice prvního řádu (First Order ODE) === | ||
| + | |||
| + | == Lineární rovnice == | ||
| + | |||
| + | <box round 90% red|**Lineární rovnice**> | ||
| + | Tvar: <m>{dy}/{dt} + p(t)y = g(t)</m> | ||
| + | |||
| + | Řešení: <m>y(t) = {\int{}{}{\mu(t)g(t)dt} + c}/{\mu(t)}</m>, <m>\mu(t) = e^{\int{}{}{p(t)dt}}</m> | ||
| + | |||
| + | Řešení2: | ||
| + | |||
| + | 1. určíme řešení rovnice <m>y\prime + p(t)y = 0</m> ve tvaru <m>y(t) = cF(t)</m> | ||
| + | |||
| + | 2. <m>y(t) = cF(t)</m> přepíšeme na <m>y(t) = c(t)F(t)</m> a dosadíme do původní rovnice | ||
| + | |||
| + | 3. spočíme c(t), dosadíme do <m>y(t) = c(t)F(t)</m> | ||
| + | |||
| + | </box> | ||
| + | |||
| + | == příklad == | ||
| + | <box round 90% green|**příklad**> | ||
| + | |||
| + | <m>v\prime = 9.8 - 0.196v</m> | ||
| + | |||
| + | upravíme do správného tvaru | ||
| + | |||
| + | <m>v\prime + 0.196v = 9.8</m> | ||
| + | |||
| + | <m>\mu(t) = e^{\int{}{}{0.196dt}} = e^{0.196t}</m> | ||
| + | |||
| + | <m>y(t) = {\int{}{}{e^{0.196t}.9.8dt} + c}/{e^{0.196t}} = {{9.8}/{0.196}e^{0.196t} + c}/{e^{0.196t}} = 50 + ce^{-0.196t}</m> | ||
| + | |||
| + | \\ | ||
| + | \\ | ||
| + | |||
| + | <m>v\prime + 0.196v = 0</m> | ||
| + | |||
| + | <m>{dv}/{dt} = -0.196v</m> | ||
| + | |||
| + | <m>{dv}/{v} = -0.196dt</m> | ||
| + | |||
| + | <m>ln|v| = -0.196t + c</m> | ||
| + | |||
| + | <m>|v| = e^{-0.196t + c}</m> | ||
| + | |||
| + | <m>|v| = e^{-0.196t}e^c</m> | ||
| + | |||
| + | <m>|v| = e^{-0.196t}c</m> | ||
| + | |||
| + | **pro kladné v**: | ||
| + | |||
| + | <m>v = ce^{-0.196t} \doubleright v = c(t)e^{-0.196t}</m> | ||
| + | |||
| + | <m>v\prime = c(t)\prime e^{-0.196t} -0.196c(t)e^{-0.196t}</m> | ||
| + | |||
| + | dosadíme: | ||
| + | |||
| + | <m>c(t)\prime e^{-0.196t} -0.196c(t)e^{-0.196t} + 0.196.c(t)e^{-0.196t} = 9.8</m> | ||
| + | |||
| + | <m>c(t)\prime e^{-0.196t} = 9.8</m> | ||
| + | |||
| + | <m>c(t)\prime = 9.8e^{+0.196t}</m> | ||
| + | |||
| + | <m>c(t) = 50e^{+0.196t} + k</m> | ||
| + | |||
| + | <m>v = c(t)e^{-0.196t} = (50e^{+0.196t} + k)e^{-0.196t} = 50 + ke^{-0.196t}</m> | ||
| + | |||
| + | **pro záporné v**: | ||
| + | |||
| + | <m>-v = ce^{-0.196t} \doubleright v = c(t)(-1)e^{-0.196t}</m> | ||
| + | |||
| + | <m>v\prime = c(t)\prime(-1)e^{-0.196t} +0.196c(t)e^{-0.196t}</m> | ||
| + | |||
| + | dosadíme: | ||
| + | |||
| + | <m>-c(t)\primee^{-0.196t} + 0.196c(t)e^{-0.196t} - 0.196.c(t)e^{-0.196t} = 9.8</m> | ||
| + | |||
| + | <m>-c(t)\prime e^{-0.196t} = 9.8</m> | ||
| + | |||
| + | <m>c(t)\prime = -9.8e^{+0.196t}</m> | ||
| + | |||
| + | <m>c(t) = -50e^{+0.196t} + k</m> | ||
| + | |||
| + | <m>v = -c(t)e^{-0.196t} = -(-50e^{+0.196t} - k)e^{-0.196t} = 50 - ke^{-0.196t} = 50 + ke^{-0.196t}</m> | ||
| + | |||
| + | </box> | ||
| + | |||
| + | == Separabilní dif. rovnice (Separable ...) == | ||
| + | |||
| + | <box round 90% red|**Separabilní dif. rovnice**> | ||
| + | Tvar: <m>N(u){dy}/{dx} = M(x)</m> | ||
| + | |||
| + | Řešení: <m>N(u){dy}/{dx} = M(x) \doubleleftright N(u)dy = M(x)dx \doubleright \int{}{}{N(u)dy} = \int{}{}{M(x)dx}</m> | ||
| + | </box> | ||
| + | |||
| + | == příklad == | ||
| + | <box round 90% green|**příklad**> | ||
| + | |||
| + | <m>y\prime = {3x^2 + 4x - 4}/{2y - 4}, y(1) = 3</m> | ||
| + | |||
| + | <m>{dy}/{dx} = {3x^2 + 4x - 4}/{2y - 4}</m> | ||
| + | |||
| + | <m>{2y - 4}{dy} = {3x^2 + 4x - 4}{dx}</m> | ||
| + | |||
| + | <m>y^2 - 4y = x^3 + 2x^2 - 4x + k</m> | ||
| + | |||
| + | partikularni reseni: | ||
| + | |||
| + | <m>9 - 12 = 1 + 2 - 4 + k</m> | ||
| + | |||
| + | <m>k = -2</m> | ||
| + | |||
| + | <m>y^2 - 4y = x^3 + 2x^2 - 4x - 2</m> | ||
| + | |||
| + | </box> | ||
| + | |||
| + | == Exact Differential Equations == | ||
| + | |||
| + | === Diferenciální rovnice druhého řádu (Second Order ODE) === | ||
| + | |||
| + | == Názvosloví == | ||
| + | |||
| + | <box round 90% blue|**Definice**> | ||
| + | **Homogenní/Nehomogenní rovnice** - <m>g(t) = 0</m>/<m>g(t) \ne 0</m> | ||
| + | </box> | ||
| + | |||
| + | <box round 90% red|**Second Order ODE, Homogenous, constant coeff.**> | ||
| + | Tvar: <m>ay\prime\prime + by\prime + cy = 0</m> | ||
| + | |||
| + | Řešení ve tvaru: <m>y(t)=e^{rt}</m> | ||
| + | |||
| + | Dosadíme: | ||
| + | |||
| + | <m>a(e^{rt})^{\prime\prime} + b(e^{rt})^\prime + ce^{rt} = 0</m> | ||
| + | |||
| + | <m>ar^2e^{rt} + bre^{rt} + ce^{rt} = 0</m> | ||
| + | |||
| + | <m>e^{rt}(ar^2 + br + c) = 0</m> | ||
| + | |||
| + | rovnice bude rovna nule, pokud bude rovno nule <m>(ar^2 + br + c) = 0</m>, r dostaneme řešením kvadratické (charakteristické rovnice) rovnice | ||
| + | |||
| + | **D > 0**: <m>y(t) = c_1e^{r_1t} + c_2e^{r_2t}</m> | ||
| + | |||
| + | **D = 0**: <m>y(t) = c_1te^{rt} + c_2e^{rt}</m> | ||
| + | |||
| + | **D < 0**: <m>y(t) = c_1e^{rt}cos(rt) + c_2e^{rt}sin(rt)</m> | ||
| + | |||
| + | </box> | ||
| + | |||
| + | <box round 90% red|**Second Order ODE, Nonhomogenous, constant coeff.**> | ||
| + | |||
| + | Určíme partikulární řešení. | ||
| + | |||
| + | **Hádáním řešení (Undetermined Coefficients)** | ||
| + | |||
| + | ^ <m>g(t)</m> ^ <m>Y_p(t)</m> guess ^ | ||
| + | | <m>ae^{\beta t}</m> | <m>Ae^{\beta t}</m> | | ||
| + | | <m>acos(\beta t)</m> | <m>Acos(\beta t) + Bsin(\beta t)</m> | | ||
| + | | <m>asin(\beta t)</m> | <m>Acos(\beta t) + Bsin(\beta t)</m> | | ||
| + | | <m>acos(\beta t) + asin(\beta t)</m> | <m>Acos(\beta t) + Bsin(\beta t)</m> | | ||
| + | | <m>3t^2 + 4t + 6</m> | <m>At^2 + Bt + C</m> | | ||
| + | |||
| + | Dopočítáme homogenní a dostaneme: <m>y(t) = Y_P(t) + Y_H(t)</m> | ||
| + | |||
| + | </box> | ||
| + | |||
| + | == příklad == | ||
| + | <box round 90% green|**příklad**> | ||
| + | |||
| + | <m>y\prime\prime + 3y\prime - 10y = 0</m> | ||
| + | |||
| + | charakteristicka rovnice: | ||
| + | |||
| + | <m>{\lambda}^2 + 3{\lambda} - 10 = 0</m> | ||
| + | |||
| + | <m>({\lambda} + 5)({\lambda} - 2) = 0</m> | ||
| + | |||
| + | <m>{\lambda_1} = 2</m> | ||
| + | |||
| + | <m>{\lambda_2} = -5</m> | ||
| + | |||
| + | reseni: | ||
| + | |||
| + | <m>y = c_1e^{2t} + c_2e^{-5t}</m> | ||
| + | |||
| + | // | ||
| + | // | ||
| + | |||
| + | <m>y\prime\prime - 4y\prime + 4y = 0</m> | ||
| + | |||
| + | charakteristicka rovnice: | ||
| + | |||
| + | <m>{\lambda}^2 - 4{\lambda} + 4 = 0</m> | ||
| + | |||
| + | <m>{({\lambda} -2)}^2 = 0</m> | ||
| + | |||
| + | <m>{\lambda} = 2</m> | ||
| + | |||
| + | reseni: | ||
| + | |||
| + | <m>y = c_1e^{2t} + c_2te^{2t}</m> | ||
| + | |||
| + | // | ||
| + | // | ||
| + | |||
| + | <m>y\prime\prime - 4y\prime + 9y = 0</m> | ||
| + | |||
| + | charakteristicka rovnice: | ||
| + | |||
| + | <m>{\lambda}^2 - 4{\lambda} + 9 = 0</m> | ||
| + | |||
| + | <m>D = 16 - 36 = - 20</m> | ||
| + | |||
| + | <m>{\lambda}_{1,2} = {4 \pm 2i\sqrt{5}}/2</m> | ||
| + | |||
| + | <m>{\lambda}_{1,2} = 2 \pm i\sqrt{5}</m> | ||
| + | |||
| + | reseni: | ||
| + | |||
| + | <m>y = c_1e^{2t}sin(\sqrt{5}t) + c_2e^{2t}cos(\sqrt{5}t)</m> | ||
| + | </box> | ||
| + | |||
| + | <box round 90% green|**příklad**> | ||
| + | |||
| + | <m>y\prime\prime - 4y\prime - 12y = 3e^{5t}</m> | ||
| + | |||
| + | 1.) partikularni reseni hadanim | ||
| + | |||
| + | <m>Y_p = Ae^{5t}</m> | ||
| + | |||
| + | <m>Y_p\prime = 5Ae^{5t}</m> | ||
| + | |||
| + | <m>Y_p\prime\prime = 25Ae^{5t}</m> | ||
| + | |||
| + | <m>25Ae^{5t} - 20Ae^{5t} - 12Ae^{5t} = 3e^{5t}</m> | ||
| + | |||
| + | <m>-7Ae^{5t} = 3e^{5t}</m> | ||
| + | |||
| + | <m>A = -3/7</m> | ||
| + | |||
| + | <m>Y_p = -3/7e^{5t}</m> | ||
| + | |||
| + | 2.) homogenní řešení | ||
| + | |||
| + | <m>{\lambda}^2 - 4{\lambda} - 12 = 0</m> | ||
| + | |||
| + | <m>(\lambda - 6)(\lambda + 2) = 0</m> | ||
| + | |||
| + | <m>\lambda_1 = 6</m> | ||
| + | |||
| + | <m>\lambda_2 = -2</m> | ||
| + | |||
| + | <m>Y_h = c_1e^{6t} + c_2e(-2t)</m> | ||
| + | |||
| + | 3.) <m>Y = Y_h + Y_p</m> | ||
| + | |||
| + | <m>Y = c_1e^{6t} + c_2e(-2t) + -3/7e^{5t}</m> | ||
| + | |||
| + | |||
| + | </box> | ||
| + | |||
| + | ===== Potvrzení ===== | ||
| + | |||
| + | <doodle single login|17a3> | ||
| + | ^ OK ^ !!! ^ | ||
| + | </doodle> | ||
| + | |||
| + | {{tag>vagabund obyc_diferencialni_rovnice linearni_rovnice}} | ||
| + | |||
| + | ~~DISCUSSION~~ | ||
